The Bonferonni and Šidák Corrections
α[PT ] =
number of significant tests total number of tests
2, 403 = .0479 . 50, 000
This value falls close to the theoretical value of α = .05.
For 7,868 families, no test reaches significance. Equivalently for 2,132 families (10,000−7,868) at least one Type I error is made. From these data, α[PF ] can be estimated as:
number of families with at least 1 Type I error α[PF ] = total number of families
2, 132 = .2132 . 10, 000
This value falls close to the theoretical value of
α[PF ] = 1 − (1 − α[PT ])C = 1 − (1 − .05)5 = .226 .
How to correct for multiple tests: Šidàk, Bonfer- onni, Boole, Dunn
Recall that the probability of making as least one Type I error for a family of C tests is
α[PF ] = 1 − (1 − α[PT ])C
This equation can be rewritten as
α[PT ] = 1 − (1 − α[PF ])1/C .
This formula—derived assuming independence of the tests—is some- times called the Šidàk equation. It shows that in order to reach a given α[PF ] level, we need to adapt the α[PT ] values used for each test.
Because the Šidàk equation involves a fractional power, it is dif- ficult to compute by hand and therefore several authors derived